$$ \hspace*{2 mm}\mathrm{1.\kern3mmx^2 − 4x\ \ge\ 5\kern2mm\ } $$ $$ \hspace*{6 mm}\mathrm{x^2 − 4x − 5\ \ge\ 0\ \ \ \ . . . gather\ all\ the\kern2mm\ } $$
$$ \hspace*{36 mm}\mathrm{terms\ on\ the\ left\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{(x − 5)(x + 1)\ \ge\ 0\ \ \ \ . . . factorise\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{x = 5\ \kern2mm\ OF\ \kern2mm\ x = −1\ \ . . . find\ the\ critical\ values\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{Now\ plot\ the\ critical\ values\ on\ a\ number\ line.\kern2mm\ } $$
There are three methods where by the
inequalities can be solved.
Study the methods and use the one
that you understand best.
We will change the method to
solve the inequality.
$$ \hspace*{28 mm}\mathrm{\bold{Method\ 1}\kern2mm\ } $$
There are two critical values, i.e. x = −1
and x = 5. The sign will change at each
critical value. Determine the sign of
(x − 5) and of (x + 1).
If x < − 1, then x + 1 < 0 and x − 5 < 0,
so that the product (x − 5)(x + 1) > 0.
If x > 5, then x − 5 > 0 and x + 1 > 0,
so that the product (x − 5)(x + 1) > 0.
Represent the data on a number line,
and determine the solution.
Solution : x ≤ − 1 OR x ≥ 5.
$$ \hspace*{28 mm}\mathrm{\bold{Method\ 2}\kern2mm\ } $$
Use the critical values, i.e. x = −1 and
x = 5 as the x-intersects of the parabola
(x − 5)(x + 1) and sketch it.
The function is positive above the x-axis and thus x ≤ − 1 or x ≥ 5
$$ \hspace*{28 mm}\mathrm{\bold{Method\ 3}\kern2mm\ } $$
Use a number line but make use of
a table, as in the diagram.
Solution : x ≤ − 1 or x ≥ 5.
[ Q 1. ]
$$ \hspace*{2 mm}\mathrm{2.\kern3mmx^2 − 3x − 4\ \le\ 0\kern2mm\ } $$ $$ \hspace*{6 mm}\mathrm{(x − 4)(x + 1)\ \le\ 0\ \ \ \ . . . factorise\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{x = 4\ \kern2mm\ OF\ \kern2mm\ x = −1\ \ . . . find\ the\ critical\ values\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{Critical\ valuess\ x = 4\ and\ x = − 1\kern2mm\ } $$
Use the critical values, i.e. x = −1 and
x = 4 as the x-intersects of the parabola
(x − 4)(x + 1). Sketch it.
The function is negative and thus the
values are below the x-axis.
Solution : − 1 ≤ x ≤ 4
[ Q 2. ]
$$ \hspace*{2 mm}\mathrm{3.\kern3mm(x − 2)(x + 3)\ >\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{critical\ values\ x = 2\ and\ x = − 3\kern2mm\ } $$
Use the tsble method.
Solution: x < − 3 or x > 2
[ Q 3. ]
$$ \hspace*{2 mm}\mathrm{4.\kern3mm(x + 1)(x − 3)\ >\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = −1\ and\ x = 3\kern2mm\ } $$
Use a parabola.
Solution: x < − 1 or x > 3
[ Q 4. ]
$$ \hspace*{2 mm}\mathrm{5.\kern3mm(x + 1)(x − 3)\ >\ 12\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{x^2 − 2x − 3 − 12\ >\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(x − 5)(x + 3)\ >\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\\ x = 5\ and\ x = − 3\kern2mm\ } $$
Use a parabola.
Solution : x < − 3 or x > 5 [ Q 5. ]
$$ \hspace*{2 mm}\mathrm{6.\kern3mm(x + 1)(x + 2)\ >\ 20\kern2mm\ } $$ $$ \hspace*{8 mm}\mathrm{x^2 + 3x − 18\ >\ 0\kern2mm\ } $$ $$ \hspace*{8 mm}\mathrm{(x − 3)(x + 6)\ >\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\\ x = 3\ and\ x = − 6\kern2mm\ } $$
Use a parabola.
Solution: x < − 6 or x > 3 [ Q 6. ]
$$ \hspace*{2 mm}\mathrm{7.\kern3mm(x − 1)(x + 4)\ \ge\ 6\kern2mm\ } $$ $$ \hspace*{8 mm}\mathrm{x^2 + 3x − 10\ \ge\ 0\kern2mm\ } $$ $$ \hspace*{8 mm}\mathrm{(x − 2)(x + 5)\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = 2\ and\ x = − 5\kern2mm\ } $$
Solution: x ≤ − 5 or x ≥ 2 [ Q 7. ]
$$ \hspace*{2 mm}\mathrm{8.\kern3mmx^2 − 9x\ \ge\ 36\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{x^2 − 9x − 36\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(x − 12)(x + 3)\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = 12\ and\ x = − 3\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{9.\kern3mmx^2 + 7x − 8\ <\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(x − 1)(x + 8)\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = 1\ and\ x = − 8\kern2mm\ } $$
Solution : − 8 < x < 1 [ Q 9. ]
$$ \hspace*{2 mm}\mathrm{10.\kern3mm7x^2 + 18x − 9\ >\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(x + 3)(7x − 3)\ >\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = − 3\ and\ x = \frac{3}{7}\kern2mm\ } $$
Solution : x < − 3 or x > 3/7 [ Q 10. ]
$$ \hspace*{2 mm}\mathrm{11.\kern3mm(2x − 3)^2\ \ge\ 169\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{4x^2 − 12x + 9 − 169\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{x^2 − 3x − 40\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(x + 5)(x − 8)\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = − 5\ en\ x = 8\kern2mm\ } $$
Solution : x < − 5 of x > 8 [ Q 11. ]
$$ \hspace*{2 mm}\mathrm{12.\kern3mmx^2 − 3x\ >\ 10\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{x^2 − 3x − 10\ >\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(x − 5)(x + 2)\ >\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = 5\ and\ x = − 2\kern2mm\ } $$
Solution : x < − 2 or x > 5 [ Q 12. ]
$$ \hspace*{2 mm}\mathrm{13.\kern3mm(x + 1)(4 − x)\ >\ 0\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{−x^2 + 3x + 4\ >\ 0\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{x^2 − 3x − 4\ \bold{<}\ 0\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{(x + 1)(x − 4)\ <\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = −1\ and\ x = 4\kern2mm\ } $$
Solution : − 1 < x < 4 [ Q 13. ]
$$ \hspace*{2 mm}\mathrm{14.\kern3mm−3(x + 7)(x − 5)\ <\ 0\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{−3x^2 − 6x + 105\ <\ 0\kern2mm\ } $$
$$ \hspace*{17 mm}\mathrm{x^2 + 2x − 35\ \bold{>}\ 0\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{(x + 7)(x − 5)\ >\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = −7\ and\ x = 5\kern2mm\ } $$
Solution : x < − 7 or x > 5 [ Q 14. ]
$$ \hspace*{2 mm}\mathrm{15.\kern3mm3 − x\ <\ 2x^2\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{2x^2 + x −3\ >\ 0\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{(2x + 3)(x − 1)\ >\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = −\frac{3}{2}\ and\ x = 1\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{16.\kern3mm4x^2 + 1\ \ge\ 5x\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{4x^2 − 5x + 1\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{(4x − 1)(x − 1)\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = \frac{1}{4}\ and\ x = 1\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{17.\kern3mmx^2 + x\ <\ 2x + 6\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{x^2 − x − 6\ <\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{(x + 2)(x − 3)\ <\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = − 2\ and\ x = 3\kern2mm\ } $$
thus the produk is positive.
If x > 3, then x + 2 > 0 and x − 3 > 0,
thus the product is positive.
Solution : − 2 < x < 3 [ Q 17. ]
$$ \hspace*{2 mm}\mathrm{18.\kern3mm2x^2 + x\ \ge\ 2x + 3\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{2x^2 − x − 3\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{(x + 1)(2x − 3)\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = − 1\ and\ x = \frac{3}{2}\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{19.\kern3mm\frac{x + 1}{x − 3}\ \le\ 2\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{x + 1}{x − 3} − 2\ \le\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{x + 1 − 2(x − 3)}{x − 3}\ \le\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{− x + 7}{x − 3}\ \le\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = 7\ and\ x = 3\kern2mm\ } $$
x ≠ 3 because you may not divide
by zero.
Solution : x < 3 or x ≥ 7 [ Q 19. ]
$$ \hspace*{2 mm}\mathrm{20.\kern3mm\frac{2x − 1}{x + 4}\ \ge\ 5\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{2x − 1}{x + 4} − 5\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{2x − 1 − 5(x + 4)}{x + 4}\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{− 3x − 21}{x + 4}\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{− 3(x + 7)}{x + 4}\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{(x + 7)}{x + 4}\ \le\ 0\ \ \dots \ divide\ by\ −3\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = − 7\ en\ x = − 4\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{21.\kern3mm\frac{3x − 1}{2x + 3}\ \le\ 4\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{3x − 1}{2x + 3} − 4\ \le\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{3x − 1 − 4(2x + 3)}{2x + 3}\ \le\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{− 5x − 13}{2x + 3}\ \le\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{5x + 13}{2x + 3}\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = −\frac{13}{5} \ and\ x = −\frac{3}{2}\kern2mm\ } $$
[ Q 21. ]
$$ \hspace*{2 mm}\mathrm{22.\kern3mm\frac{1 − 2x}{x + 3}\ \le\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x =\frac{1}{2} \ and\ x = − 3\kern2mm\ } $$
[ Q 22. ]
$$ \hspace*{2 mm}\mathrm{23.\kern3mm\frac{5 − 2x}{x + 6}\ \ge\ 1\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{5 − 2x}{x + 6} − 1\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{5 − 2x − (x + 6)}{x + 6}\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{− 3x − 1}{x + 6}\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{3x + 1}{x + 6}\ \le\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = −\frac{1}{3} \ and\ x = − 6\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{24.\kern3mm\frac{1 − 3x}{x − 4}\ \le\ 3\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{1 − 3x}{x − 4} − 3\ \le\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{1 − 3x − 3(x − 4)}{x − 4}\ \le\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{− 6x + 13}{x − 4}\ \le\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{6x − 13}{x − 4}\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = \frac{13}{6} \ and\ x = 4\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{25.\kern3mm\frac{x^2 + 4x − 6}{2x + 3}\ \le\ 3\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{x^2 + 4x − 6}{2x + 3} − 3\ \le\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{x^2 + 4x − 6 − 3(2x + 3)}{2x + 3}\ \le\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{x^2 − 2x − 15}{2x + 3}\ \le\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{(x + 3)(x − 5)}{2x + 3}\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = − 3 \ or\ x = − \frac{3}{2}\ or\ x = 5\kern2mm\ } $$
[ Q 25. ]
$$ \hspace*{2 mm}\mathrm{26.\kern3mm\frac{x^2 − 2x + 4}{2x − 5}\ \le\ 7\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{x^2 − 2x + 4}{2x − 5} − 7\ \le\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{x^2 − 2x + 4 − 7(2x − 5)}{2x − 5}\ \le\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{x^2 − 16x + 39}{2x − 5}\ \le\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{(x − 3)(x − 13)}{2x − 5}\ \le\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = \frac{5}{2}\ \ or\ \ x = 3\ \ or\ \ x = 13\kern2mm\ } $$
[ Q 26. ]
$$ \hspace*{2 mm}\mathrm{27.\kern3mm\frac{x^2 + 3x − 1}{x + 1}\ \ge\ 3\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{x^2 + 3x − 1}{x + 1} − 3\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{x^2 + 3x − 1 − 3(x + 1)}{x + 1}\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{x^2 − 4}{x + 1}\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{\frac{(x − 2)(x + 2)}{x + 1}\ \ge\ 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Critical\ values\ x = − 3\ \ or\ \ x = − 1\ \ or\ \ x = 2\kern2mm\ } $$
[ Q 27. ]